A circle on the Coordinate plane has a diameter with endpoints at (6,8) and (15,8). What are the coordinates of the center of the circle? What is the radius of the circle? Identify the coordinates of another point on the circle. Explain how you found your answer.
Well, I hope this information doesn't go in one ear and out the other, but let's try to solve this problem!
The center of a circle is located right in the middle, which is about the only place you'll find me without my red nose. To find the center, we can take the average of the x-coordinates and the y-coordinates of the two endpoints.
The average of the x-coordinates is (6 + 15) / 2 = 21 / 2 = 10.5.
The average of the y-coordinates is (8 + 8) / 2 = 16 / 2 = 8.
So, the coordinates of the center are (10.5, 8). Voila!
Now, let's move on to the radius. The distance between the center and any point on the circle is what we call the radius.
Since the endpoints are (6,8) and (15,8), it's pretty clear that the radius is half of the distance between them. In this case, it's (15 - 6) / 2 = 9 / 2 = 4.5.
Therefore, the radius of the circle is 4.5. I'm not clowning around with you!
To find another point on the circle, we can go in any direction from the center by the length of the radius. For example, let's go to the right. Starting from the center (10.5, 8), if we move 4.5 units to the right, we get (15, 8), one of the endpoints of the diameter.
So, the coordinates of another point on the circle are (15, 8).
I hope that wasn't too much of a juggling act for you!
To find the center of the circle, we first need to find the average of the x-coordinates and the average of the y-coordinates of the endpoints of the diameter.
The x-coordinate of the center is the average of the x-coordinates of the endpoints:
Average x-coordinate = (6 + 15) / 2 = 21 / 2 = 10.5
The y-coordinate of the center is the average of the y-coordinates of the endpoints:
Average y-coordinate = 8
Therefore, the coordinates of the center of the circle are (10.5, 8).
To find the radius, we can use the distance formula between the center of the circle and one of the endpoints of the diameter. Let's use (6, 8):
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Distance = √((10.5 - 6)^2 + (8 - 8)^2)
Distance = √(4.5^2 + 0^2)
Distance = √(20.25)
Distance = 4.5
Therefore, the radius of the circle is 4.5 units.
To find the coordinates of another point on the circle, we can use the fact that any point on the circle will be a distance of 4.5 units away from the center.
Let's choose a point to the right of the center. To get the x-coordinate, we can add 4.5 units to the x-coordinate of the center:
x-coordinate = 10.5 + 4.5 = 15
The y-coordinate will remain the same, since the point will lie on the same horizontal line:
y-coordinate = 8
Therefore, another point on the circle is (15, 8).
To find the coordinates of the center of the circle, we can find the midpoint of the diameter. The midpoint formula states that the coordinates of the midpoint of a line segment can be found by taking the average of the x-coordinates and the average of the y-coordinates of the endpoints.
1. Average of x-coordinates: (6 + 15) / 2 = 21 / 2 = 10.5
2. Average of y-coordinates: (8 + 8) / 2 = 16 / 2 = 8
Therefore, the center of the circle has coordinates (10.5, 8).
To find the radius of the circle, we can use the distance formula. The distance formula calculates the distance between two points given their coordinates. For two points (x1, y1) and (x2, y2), the distance, d, is given by the formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2).
In this case, we can find the distance between the two endpoints of the diameter, which will give us the diameter of the circle. Then we can divide the diameter by 2 to get the radius.
1. Calculate the distance between the endpoints:
d = sqrt((15 - 6)^2 + (8 - 8)^2)
= sqrt(9^2 + 0^2)
= sqrt(81 + 0)
= sqrt(81)
= 9
2. Divide the diameter by 2 to get the radius:
radius = 9 / 2 = 4.5
Therefore, the radius of the circle is 4.5.
To find the coordinates of another point on the circle, we can use the fact that any point on a circle is equidistant from its center. Since we already know the center (10.5, 8) and the radius (4.5), we can use the distance formula to find the coordinates of another point.
1. Choose an arbitrary angle, let's say 30 degrees.
2. Convert the angle to radians: 30 degrees * (π / 180 degrees) ≈ 0.52 radians.
3. Use polar coordinates conversion formula:
x = center_x + radius * cos(angle)
y = center_y + radius * sin(angle)
= 10.5 + 4.5 * cos(0.52)
= 10.5 + 4.5 * 0.8746
≈ 14.36
= 8 + 4.5 * sin(0.52)
= 8 + 4.5 * 0.4848
≈ 10.18
Therefore, another point on the circle would have coordinates approximately (14.36, 10.18).
By using the midpoint formula, distance formula, and the polar coordinates conversion formula, we were able to find the coordinates of the center of the circle, the radius, and another point on the circle.
midpoint of diameter
= ( (6+15)/2 , (8+8)/2)
= ( 21/2 , 8)
equation:
(x-21/2)^2 + (y-8)^2 = r^2
plug in (6,8)
81/4 + 0 = r^2
(x-21/2)^2 + (y-8)^2 = 81/4
any other point?
how about the y-intercept, let x = 0
441/4 + (y-8)^2 = 81/4
y-8 = ±√90 = ±3√10
y = 8 ±√10
so (0, 8+3√10) and (0,8-3√10) are two more points